This document contains information about geometry concepts including:
- Five axioms of Euclidean geometry including belonging, ordering, congruence, parallel lines, and continuity.
- Definitions and properties of lines, angles, triangles, and parallel lines. Key theorems like Thales' theorem are also discussed.
- Explanations of angle measures in degrees and radians. Classifications of angles as acute, obtuse, right, supplementary and vertical.
- Equations for different types of lines including those passing through two points or having a given slope.
This document contains 17 multi-part math problems related to straight lines and graphs. The problems cover topics such as finding the gradient and equation of a line, determining if two lines are parallel or perpendicular, finding the midpoint and equation of a line through two given points, and interpreting the gradient of a graph. Several questions provide diagrams or graphs to accompany the problems. The document also provides attribution information, noting the exam boards and sample assessment materials that the questions were retrieved from.
This document discusses triangles and similarity. It provides examples of using the AAA, SAS, and SSS similarity criteria to determine if two triangles are similar. It also contains exercises involving applying similarity rules to pairs of triangles and finding missing angle measures. One exercise involves showing two triangles are similar using the fact that corresponding angles are equal for diagonals intersecting in a trapezium where the bases are parallel. The summary is provided in 3 sentences or less as requested.
This document discusses triangles and similarity. It provides examples of using the AAA, SAS, and SSS similarity criteria to determine if two triangles are similar. It also contains exercises involving applying similarity rules to pairs of triangles and finding missing angle measures. One exercise involves showing two triangles are similar using the fact that corresponding angles are equal for diagonals intersecting in a trapezium where the bases are parallel. The summary is provided in 3 sentences or less as requested.
1. The document provides definitions and theorems related to geometric shapes including triangles, trapezoids, kites, and parallelograms. It defines key terms like midline, median, diagonal, and congruent sides.
2. Theorems covered include the triangle midline theorem stating the midline is parallel to the third side and half its length. For trapezoids, the midsegment theorem states the median is parallel to the bases and half their sum.
3. Theorems for kites include the diagonals being perpendicular and one diagonal bisecting the other or bisecting a pair of opposite angles.
The document is an acknowledgement from a group of 5 students - Abhishek Mahto, Lakshya Kumar, Mohan Kumar, Ritik Kumar, and Vivek Singh of class X E. They are thanking their principal Dr. S.V. Sharma and math teacher Mrs. Shweta Bhati for their guidance and support in completing their project on triangles and similarity. They also thank their parents and group members for their advice and assistance during the project.
The document defines and discusses parallel and perpendicular lines. Parallel lines are always the same distance apart and never intersect. Perpendicular lines intersect to form right angles of 90 degrees. The document provides examples of finding whether two lines are parallel or perpendicular based on their angles and relationship to a transversal or third line. It also gives examples of using corresponding, alternate interior, and alternate exterior angles to determine if lines are parallel when cut by a transversal.
Parallel And Perpendicular With Real Life ExamplesToufiq Elahi
Parallel Lines :
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.
Perpendicular Lines :
Perpendicular means "at right angles". A line meeting another at a right angle, or 90° is said to be perpendicular to it.
Syllabus of jamia millia islamia; class 11 science 2021-22MD PERVEZ KHAN
This document provides the syllabus for the 11th grade entrance exam for the science stream at Jamia Millia Islamia. It outlines the subjects that will be covered on the exam, including English, General Knowledge, Physics, Chemistry, Biology, and Mathematics. For each subject, it lists the key topics that will be included, such as types of chemical reactions in Chemistry, life processes and reproduction in Biology, and reflection and refraction of light in Physics. It also provides details on the exam format, noting it will consist of 100 multiple choice questions testing knowledge of concepts from the NCERT syllabus for each class 11 subject.
This document contains 17 multi-part math problems related to straight lines and graphs. The problems cover topics such as finding the gradient and equation of a line, determining if two lines are parallel or perpendicular, finding the midpoint and equation of a line through two given points, and interpreting the gradient of a graph. Several questions provide diagrams or graphs to accompany the problems. The document also provides attribution information, noting the exam boards and sample assessment materials that the questions were retrieved from.
This document discusses triangles and similarity. It provides examples of using the AAA, SAS, and SSS similarity criteria to determine if two triangles are similar. It also contains exercises involving applying similarity rules to pairs of triangles and finding missing angle measures. One exercise involves showing two triangles are similar using the fact that corresponding angles are equal for diagonals intersecting in a trapezium where the bases are parallel. The summary is provided in 3 sentences or less as requested.
This document discusses triangles and similarity. It provides examples of using the AAA, SAS, and SSS similarity criteria to determine if two triangles are similar. It also contains exercises involving applying similarity rules to pairs of triangles and finding missing angle measures. One exercise involves showing two triangles are similar using the fact that corresponding angles are equal for diagonals intersecting in a trapezium where the bases are parallel. The summary is provided in 3 sentences or less as requested.
1. The document provides definitions and theorems related to geometric shapes including triangles, trapezoids, kites, and parallelograms. It defines key terms like midline, median, diagonal, and congruent sides.
2. Theorems covered include the triangle midline theorem stating the midline is parallel to the third side and half its length. For trapezoids, the midsegment theorem states the median is parallel to the bases and half their sum.
3. Theorems for kites include the diagonals being perpendicular and one diagonal bisecting the other or bisecting a pair of opposite angles.
The document is an acknowledgement from a group of 5 students - Abhishek Mahto, Lakshya Kumar, Mohan Kumar, Ritik Kumar, and Vivek Singh of class X E. They are thanking their principal Dr. S.V. Sharma and math teacher Mrs. Shweta Bhati for their guidance and support in completing their project on triangles and similarity. They also thank their parents and group members for their advice and assistance during the project.
The document defines and discusses parallel and perpendicular lines. Parallel lines are always the same distance apart and never intersect. Perpendicular lines intersect to form right angles of 90 degrees. The document provides examples of finding whether two lines are parallel or perpendicular based on their angles and relationship to a transversal or third line. It also gives examples of using corresponding, alternate interior, and alternate exterior angles to determine if lines are parallel when cut by a transversal.
Parallel And Perpendicular With Real Life ExamplesToufiq Elahi
Parallel Lines :
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.
Perpendicular Lines :
Perpendicular means "at right angles". A line meeting another at a right angle, or 90° is said to be perpendicular to it.
Syllabus of jamia millia islamia; class 11 science 2021-22MD PERVEZ KHAN
This document provides the syllabus for the 11th grade entrance exam for the science stream at Jamia Millia Islamia. It outlines the subjects that will be covered on the exam, including English, General Knowledge, Physics, Chemistry, Biology, and Mathematics. For each subject, it lists the key topics that will be included, such as types of chemical reactions in Chemistry, life processes and reproduction in Biology, and reflection and refraction of light in Physics. It also provides details on the exam format, noting it will consist of 100 multiple choice questions testing knowledge of concepts from the NCERT syllabus for each class 11 subject.
This document presents information about straight lines. It begins with an introduction to straight lines, defining them as lines with no curves that extend infinitely in both directions. It then discusses the Cartesian coordinate system and how points on a plane are located using x and y coordinates. Various applications of straight lines in areas like roads, physics, and data analysis are provided. Key concepts covered include properties of straight lines, parallel lines, and the relationship between slope and the angle of a line.
03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pp
1. The document defines and explains the properties of parallelograms, rectangles, rhombi, squares, and trapezoids.
2. Key properties discussed include opposite sides being parallel and congruent, opposite angles being congruent, consecutive angles being supplementary, and diagonals bisecting each other.
3. Examples are provided to demonstrate applying the properties to determine missing angle measures and side lengths.
Class 10 Cbse Maths 2010 Sample Paper Model 1Sunaina Rawat
The document provides information on the design of a mathematics question paper for Class X. It specifies:
1) The weightage and distribution of marks for different content units and forms of questions. Number systems, algebra and geometry make up the bulk of the paper.
2) The paper will contain very short answer questions, short answer questions and long answer questions.
3) Internal choices will be provided for some questions.
4) Questions will be of easy, average and difficult levels.
5) A sample question paper and marking scheme are included based on this design.
1. A parallelogram is a quadrilateral with two properties: opposite sides are parallel and opposite angles are congruent.
2. There are several types of parallelograms including rectangles, squares, and rhombi.
3. Rectangles have four right angles in addition to the properties of parallelograms. Squares have four congruent sides and four right angles. Rhombi have four congruent sides.
Engineering, Agriculture and Medical Common Entrance Test (EAMCET) is conducted by Jawaharlal Nehru Technological University Hyderabad on behalf of APSCHE. This examination is the prerequisite for admission into various professional courses offered in University/ Private Colleges in the state of Andhra Pradesh.
EAMCET 2015 is going to be conducted during May 2015 The Syllabus is based on 10+2 exam system covering class 11 & 12 syllabus.
Syllabus Contents:
1. Physics
2. Chemistry
3. Mathematics
www.entranceindia.com provides model papers for EAMCET medical and engineering entrance examinations. For more information please visit our site (www.entranceindia.com) today.
The document contains information about triangles, including:
1) If two triangles have proportional sides and equal angles, they are similar triangles.
2) In a right triangle, a perpendicular line from the right angle to the hypotenuse divides it into two right triangles that are similar to each other and to the original triangle.
3) A line dividing two sides of a triangle proportionally is parallel to the third side.
This document provides information about congruent triangles. It defines congruent triangles as two triangles that have the same shape and size, with corresponding sides and angles being equal. It describes several triangle congruence theorems including SSS, SAS, ASA, AAS, and RHS, which establish that triangles are congruent if certain combinations of sides and/or angles are equal. It also discusses isosceles triangles, angle bisectors, and provides examples applying the congruence theorems to prove triangles are congruent or not.
The document discusses solving simultaneous equations through various methods. It begins by providing context about the history of simultaneous equations and important definitions. It then focuses on 6 key techniques: 1) using properties of parallel lines to determine the number of solutions, 2) graphing the lines to find their point of intersection, 3) using substitution and elimination to solve algebraically, 4) applying matrix operations like Gauss-Jordan elimination, 5) using Cramer's rule with determinants, and 6) examples of real-world applications. The student had correctly identified that the lines in the given system of equations were parallel based on having the same slope, indicating no solution.
The document discusses proportion and similar triangles in geometry. It defines proportion as an equation stating that two ratios are equal, and provides examples of using cross products to check for proportion. It then defines similar polygons and triangles as those with congruent corresponding angles and proportional corresponding sides. The document provides different methods to prove triangles are similar, including SAS, SSS, and AA similarity. It also discusses how corresponding parts of similar triangles, such as perimeters, altitudes, angle bisectors, and medians are proportional. Several theorems and examples involving parallel lines cutting across triangles proportionally are presented.
This document provides teaching materials on geometry topics related to angles, parallel lines, and transversals. It includes definitions of terms like complementary angles, supplementary angles, and linear pairs. There are examples of problems involving finding angle measures using properties of parallel lines cut by a transversal, as well as practice problems for students to work through. The document aims to help students learn to identify and measure different types of angles formed when lines intersect.
Compiled and solved problems in geometry and trigonometry,F.SmaradanheΘανάσης Δρούγας
This document contains 29 geometry problems from Romanian textbooks for 9th and 10th grade students, compiled and solved by Florentin Smarandache. The problems cover a range of topics in geometry including properties of angles, triangles, polygons, circles, areas, and constructions. Smarandache compiled these problems during his time teaching mathematics in Romania and Morocco between 1981-1988. He provides solutions to each problem at the end of the document to serve as an educational aid for mathematics students and instructors.
Physics engineering-first-year notes, books, e book pdf downloadVinnie Singh
The document provides information about Studynama.com, an online education hub that provides free study materials for students from classes 9-12, engineers, managers, lawyers, and doctors. It includes lecture notes, project reports, and solved papers for various subjects. Disclaimers are provided noting that Studynama.com does not own the copyright for this content. The remainder of the document is a study material on elastodynamics that covers topics like simple harmonic motion, displacement and area vectors, Coulomb's law, and more.
The document discusses different types of proofs in geometry using concepts like congruence, similarity, triangles, and parallel lines. It provides examples and definitions of key terms and asks multiple choice questions to test understanding. To prove congruence, one must observe that all internal angles are the same and all angles and sides correspond. Similar shapes differ in that their lengths may be different, while congruent shapes have exactly matching side lengths and angles.
This document discusses various metric relationships in mathematics including proofs, congruency, similarity, angles, triangles, and the Pythagorean theorem. It defines key concepts such as complementary angles, supplementary angles, opposite angles, interior angles, corresponding angles, and more. Examples of exam questions are provided testing these concepts along with activities for further practice.
This document provides information about Module 5 on quadrilaterals, including:
1) An introduction focusing on identifying quadrilaterals that are parallelograms and determining the conditions for a quadrilateral to be a parallelogram.
2) A module map outlining the key topics to be covered, including parallelograms, rectangles, trapezoids, kites, and solving real-life problems.
3) A pre-assessment to gauge the learner's existing knowledge of quadrilaterals through multiple choice and short answer questions.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
This document presents information about straight lines. It begins with an introduction to straight lines, defining them as lines with no curves that extend infinitely in both directions. It then discusses the Cartesian coordinate system and how points on a plane are located using x and y coordinates. Various applications of straight lines in areas like roads, physics, and data analysis are provided. Key concepts covered include properties of straight lines, parallel lines, and the relationship between slope and the angle of a line.
03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pp
1. The document defines and explains the properties of parallelograms, rectangles, rhombi, squares, and trapezoids.
2. Key properties discussed include opposite sides being parallel and congruent, opposite angles being congruent, consecutive angles being supplementary, and diagonals bisecting each other.
3. Examples are provided to demonstrate applying the properties to determine missing angle measures and side lengths.
Class 10 Cbse Maths 2010 Sample Paper Model 1Sunaina Rawat
The document provides information on the design of a mathematics question paper for Class X. It specifies:
1) The weightage and distribution of marks for different content units and forms of questions. Number systems, algebra and geometry make up the bulk of the paper.
2) The paper will contain very short answer questions, short answer questions and long answer questions.
3) Internal choices will be provided for some questions.
4) Questions will be of easy, average and difficult levels.
5) A sample question paper and marking scheme are included based on this design.
1. A parallelogram is a quadrilateral with two properties: opposite sides are parallel and opposite angles are congruent.
2. There are several types of parallelograms including rectangles, squares, and rhombi.
3. Rectangles have four right angles in addition to the properties of parallelograms. Squares have four congruent sides and four right angles. Rhombi have four congruent sides.
Engineering, Agriculture and Medical Common Entrance Test (EAMCET) is conducted by Jawaharlal Nehru Technological University Hyderabad on behalf of APSCHE. This examination is the prerequisite for admission into various professional courses offered in University/ Private Colleges in the state of Andhra Pradesh.
EAMCET 2015 is going to be conducted during May 2015 The Syllabus is based on 10+2 exam system covering class 11 & 12 syllabus.
Syllabus Contents:
1. Physics
2. Chemistry
3. Mathematics
www.entranceindia.com provides model papers for EAMCET medical and engineering entrance examinations. For more information please visit our site (www.entranceindia.com) today.
The document contains information about triangles, including:
1) If two triangles have proportional sides and equal angles, they are similar triangles.
2) In a right triangle, a perpendicular line from the right angle to the hypotenuse divides it into two right triangles that are similar to each other and to the original triangle.
3) A line dividing two sides of a triangle proportionally is parallel to the third side.
This document provides information about congruent triangles. It defines congruent triangles as two triangles that have the same shape and size, with corresponding sides and angles being equal. It describes several triangle congruence theorems including SSS, SAS, ASA, AAS, and RHS, which establish that triangles are congruent if certain combinations of sides and/or angles are equal. It also discusses isosceles triangles, angle bisectors, and provides examples applying the congruence theorems to prove triangles are congruent or not.
The document discusses solving simultaneous equations through various methods. It begins by providing context about the history of simultaneous equations and important definitions. It then focuses on 6 key techniques: 1) using properties of parallel lines to determine the number of solutions, 2) graphing the lines to find their point of intersection, 3) using substitution and elimination to solve algebraically, 4) applying matrix operations like Gauss-Jordan elimination, 5) using Cramer's rule with determinants, and 6) examples of real-world applications. The student had correctly identified that the lines in the given system of equations were parallel based on having the same slope, indicating no solution.
The document discusses proportion and similar triangles in geometry. It defines proportion as an equation stating that two ratios are equal, and provides examples of using cross products to check for proportion. It then defines similar polygons and triangles as those with congruent corresponding angles and proportional corresponding sides. The document provides different methods to prove triangles are similar, including SAS, SSS, and AA similarity. It also discusses how corresponding parts of similar triangles, such as perimeters, altitudes, angle bisectors, and medians are proportional. Several theorems and examples involving parallel lines cutting across triangles proportionally are presented.
This document provides teaching materials on geometry topics related to angles, parallel lines, and transversals. It includes definitions of terms like complementary angles, supplementary angles, and linear pairs. There are examples of problems involving finding angle measures using properties of parallel lines cut by a transversal, as well as practice problems for students to work through. The document aims to help students learn to identify and measure different types of angles formed when lines intersect.
Compiled and solved problems in geometry and trigonometry,F.SmaradanheΘανάσης Δρούγας
This document contains 29 geometry problems from Romanian textbooks for 9th and 10th grade students, compiled and solved by Florentin Smarandache. The problems cover a range of topics in geometry including properties of angles, triangles, polygons, circles, areas, and constructions. Smarandache compiled these problems during his time teaching mathematics in Romania and Morocco between 1981-1988. He provides solutions to each problem at the end of the document to serve as an educational aid for mathematics students and instructors.
Physics engineering-first-year notes, books, e book pdf downloadVinnie Singh
The document provides information about Studynama.com, an online education hub that provides free study materials for students from classes 9-12, engineers, managers, lawyers, and doctors. It includes lecture notes, project reports, and solved papers for various subjects. Disclaimers are provided noting that Studynama.com does not own the copyright for this content. The remainder of the document is a study material on elastodynamics that covers topics like simple harmonic motion, displacement and area vectors, Coulomb's law, and more.
The document discusses different types of proofs in geometry using concepts like congruence, similarity, triangles, and parallel lines. It provides examples and definitions of key terms and asks multiple choice questions to test understanding. To prove congruence, one must observe that all internal angles are the same and all angles and sides correspond. Similar shapes differ in that their lengths may be different, while congruent shapes have exactly matching side lengths and angles.
This document discusses various metric relationships in mathematics including proofs, congruency, similarity, angles, triangles, and the Pythagorean theorem. It defines key concepts such as complementary angles, supplementary angles, opposite angles, interior angles, corresponding angles, and more. Examples of exam questions are provided testing these concepts along with activities for further practice.
This document provides information about Module 5 on quadrilaterals, including:
1) An introduction focusing on identifying quadrilaterals that are parallelograms and determining the conditions for a quadrilateral to be a parallelogram.
2) A module map outlining the key topics to be covered, including parallelograms, rectangles, trapezoids, kites, and solving real-life problems.
3) A pre-assessment to gauge the learner's existing knowledge of quadrilaterals through multiple choice and short answer questions.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
The chapter Lifelines of National Economy in Class 10 Geography focuses on the various modes of transportation and communication that play a vital role in the economic development of a country. These lifelines are crucial for the movement of goods, services, and people, thereby connecting different regions and promoting economic activities.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
Math-Readings-3.pdf
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Math III - 1 -
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GEOMETRY POINTERS
Euclidean geometry axioms
As we have noted above, there is a set of the axioms – properties, that are considered in geometry as main ones
and are adopted without a proof . Now, after introducing some initial notions and definitions we can consider
the following sufficient set of the axioms, usually used in plane geometry.
Axiom of belonging. Through any two points in a plane it is possible to draw a straight line, and besides only
one.
Axiom of ordering. Among any three points placed in a straight line, there is no more than one point placed
between the two others.
Axiom of congruence ( equality ) of segments and angles. If two segments (angles) are congruent to the third
one, then they are congruent to each other.
Axiom of parallel straight lines. Through any point placed outside of a straight line it is possible to draw
another straight line, parallel to the given line, and besides only one.
Axiom of continuity ( Archimedean axiom ). Let AB and CD be two some segments; then there is a finite set
of such points A1 , A2 , … , An , placed in the straight line AB, that segments AA1 , A1A2 , … , An - 1An are
congruent to segment CD, and point B is placed between A and An .
We emphasize, that replacing one of these axioms by another, turns this axiom into a theorem, requiring a
proof. So, instead of the axiom of parallel straight lines we can use as an axiom the property of triangle angles
(“the sum of triangle angles is equal to 180 deg”). But then we should to prove the property of parallel lines.
Straight line
A general equation of straight line:
Ах + Ву + С = 0 ,
where А and В aren't equal to zero simultaneously.
2. Academic
Academic
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-Clinic.com
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Coefficients А and В are coordinates of normal vector of the straight line ( i.e. vector, perpendicular to the
straight line ). At А = 0 straight line is parallel to the axis ОХ , at В = 0 straight line is parallel to the axis ОY .
At В 0 we receive an equation of straight line with a slope:
An equation of the straight line, going through the point ( х0 , у 0 ) and not parallel to the axis OY :
у – у 0 = m ( x – х0 ) ,
where m is a slope, equal to tangent of an angle between the straight line and the positive direction of the axis
ОХ .
At А 0, В 0 and С 0 we receive an equation of straight line in segments on axes:
where a = – C / A, b = – C / B. This line goes through the points ( a, 0 ) and ( 0, b ), i.e. it cuts off segments a
and b long on the coordinate axes.
An equation of straight line going through two different points ( х1, у 1 ) and ( х2, у 2 ):
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Parallel straight lines
Two straight lines AB and CD ( Fig.11 ) are called parallel straight lines, if they lie in the same plane and don’t
intersect however long they may be continued. The designation: AB|| CD. All points of one line are equidistant
from another line. All straight lines, parallel to one straight line are parallel between themselves. It’s adopted
that an angle between parallel straight lines is equal to zero. An angle between two parallel rays is equal to zero,
if their directions are the same and 180 deg, if the directions are opposite. All perpendiculars (AB, CD, EF, and
Fig.12) to the one straight line KM are parallel between themselves. Inversely, the straight line KM, which is
perpendicular to one of parallel straight lines, is perpendicular to all others. A length of perpendicular segment,
concluded between two parallel straight lines, is a distance between them.
At intersecting two parallel straight lines by the third line, eight angles are formed (Fig.13), which are called
two-by-two:
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1) corresponding angles (1 and 5; 2 and 6; 3 and 7; 4 and 8 ); these angles
are equal two-by-two: ( 1 = 5; 2 = 6; 3 = 7; 4 = 8 );
2) alternate interior angles ( 4 and 5; 3 and 6 ); they are equal two-by-two;
3) alternate exterior angles ( 1 and 8; 2 and 7 ); they are equal two-by-two;
4) one-sided interior angles (3 and 5; 4 and 6 ); a sum of them two-by-two
is equal to180 deg ( 3 + 5 = 180 deg; 4 + 6 = 180 deg);
5) one-sided exterior angles ( 1 and 7; 2 and 8 ); a sum of them two-by-two
is equal to180 deg ( 1 + 7 = 180 deg; 2 + 8 = 180 deg).
Angles with correspondingly parallel sides either are equal one to another, ( if both of them are acute or both
are obtuse, 1 = 2, Fig.14 ), or sum of them is 180 deg ( 3 + 4 = 180 deg, Fig.15 ).
Angles with correspondingly perpendicular sides are also either equal one to another ( if both of them are acute
or both are obtuse ), or sum of them is 180 deg.
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Thales' theorem. At intersecting sides of an angle by parallel lines ( Fig.16 ), the angle sides are divided into
the proportional segments:
Angles
Angle is a geometric figure ( Fig.1 ), formed by two rays OA and OB ( sides of an angle ), going out of the
same point O (a vertex of an angle).
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An angle is signed by the symbol and three letters, marking ends of rays and a vertex of an angle: AOB
(moreover, a vertex letter is placed in the middle). A measure of an angle is a value of a turn around a vertex O,
that transfers a ray OA to the position OB. Two units of angles measures are widely used: a radian and a
degree. About a radian measure see below in the point “A length of arc” and also in the section
“Trigonometry”.
A degree measure. Here a unit of measurement is a degree ( its designation is ° or deg ) – a turn of a ray by the
1/360 part of the one complete revolution. So, the complete revolution of a ray is equal to 360 deg. One degree
is divided by 60 minutes ( a designation is ‘ or min ); one minute – correspondingly by 60 seconds ( a
designation is “ or sec ). An angle of 90 deg ( Fig.2 ) is called a right or direct angle; an angle lesser than 90
deg ( Fig.3 ), is called an acute angle; an angle greater than 90 deg ( Fig.4 ), is called an obtuse angle.
Straight lines, forming a right angle, are called mutually perpendicular lines. If the straight lines AB and MK
are perpendicular, this is signed as: AB MK.
Signs of angles. An angle is considered as positive, if a rotation is executed opposite a clockwise, and negative –
otherwise. For example, if the ray OA displaces to the ray OB as shown on Fig.2, then AOB = + 90 deg; but
on Fig.5 AOB = – 90 deg.
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Supplementary (adjacent) angles ( Fig.6 ) – angles AOB and COB, having the common vertex O and the
common side OB; other two sides OA and OC form a continuation one to another. So, a sum of supplementary
(adjacent) angles is equal to 180 deg.
Vertically opposite (vertical) angles ( Fig.7) – such two angles with a common vertex, that sides of one angle
are continuations of the other: AOB and COD ( and also AOC and DOB ) are vertical angles.
A bisector of an angle is a ray, dividing the angle in two (Fig.8). Bisectors of vertical angles (OM and ON,
Fig.9) are continuations one of the other. Bisectors of supplementary angles (OM and ON, Fig.10) are mutually
perpendicular lines.
The property of an angle bisector: any point of an angle bisector is placed by the same distance from the angle
sides.
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Triangle
Triangle is a polygon with three sides (or three angles). Sides of triangle are signed often by small letters,
corresponding to designations of opposite vertices, signed by capital letters.
If all the three angles are acute ( Fig.20 ), then this triangle is an acute-angled triangle; if one of the angles is
right ( C, Fig.21 ), then this triangle is a right-angled triangle; sides a, b, forming a right angle, are called
legs; side c, opposite to a right angle, called a hypotenuse; if one of the angles is obtuse ( B, Fig.22 ), then
this triangle is an obtuse-angled triangle.
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A triangle ABC is an isosceles triangle (Fig.23), if the two of its sides are equal (a = c); these equal sides are
called lateral sides, the third side is called a base of triangle. A triangle ABC is an equilateral triangle (Fig.24),
if all of its sides are equal
(a = b = c). In general case ( a b c ) we have a scalene triangle.
Main properties of triangles. In any triangle:
1. An angle, lying opposite the greatest side, is also the greatest angle, and inversely.
2. Angles, lying opposite the equal sides, are also equal, and inversely. In particular,
all angles in an equilateral triangle are also equal.
3. A sum of triangle angles is equal to 180 deg.
From the two last properties it follows, that each angle in an equilateral triangle
is equal to 60 deg.
4. Continuing one of the triangle sides (AC , Fig. 25), we receive an exterior angle BCD.
An exterior angle of a triangle is equal to a sum of interior angles, not supplementary
with it: BCD = A + B.
5. Any side of a triangle is less than a sum of two other sides and more than their
difference ( a < b + c, a > b – c; b < a + c, b > a – c; c < a + b, c > a – b ).
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Theorems about congruence of triangles.
Two triangles are congruent, if they have accordingly equal:
a) two sides and an angle between them;
b) two angles and a side, adjacent to them;
c) three sides.
Theorems about congruence of right-angled triangles.
Two right-angled triangles are congruent, if one of the following conditions is valid:
1) their legs are equal;
2) a leg and a hypotenuse of one of triangles are equal to a leg and a hypotenuse of another;
3) a hypotenuse and an acute angle of one of triangles are equal to a hypotenuse and
an acute angle of another;
4) a leg and an adjacent acute angle of one of triangles are equal to a leg and an
adjacent acute angle of another;
5) a leg and an opposite acute angle of one of triangles are equal to a leg and an opposite acute angle of
another.
Remarkable lines and points of triangle.
Altitude (height) of a triangle is a perpendicular, dropped from any vertex to an opposite side (or to its
continuation). This side is called a base of triangle in this case. Three heights of triangle always intersect in one
point, called an orthocenter of a triangle. An orthocenter of an acute-angled triangle (point O, Fig.26) is placed
inside of the triangle; and an orthocenter of an obtuse-angled triangle (point O, Fig.27) – outside of the triangle;
an orthocenter of a right-angled triangle coincides with a vertex of the right angle.
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Median is a segment, joining any vertex of triangle and a midpoint of the opposite side. Three medians of
triangle ( AD, BE, CF, Fig.28 ) intersect in one point O (always lied inside of a triangle), which is a center of
gravity of this triangle. This point divides each median by ratio 2:1, considering from a vertex.
Bisector is a segment of the angle bisector, from a vertex to a point of intersection with an opposite side. Three
bisectors of a triangle (AD, BE, CF, Fig.29) intersect in the one point (always lied inside of triangle), which is
a center of an inscribed circle (see the section “Inscribed and circumscribed polygons”).
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A bisector divides an opposite side into two parts, proportional to the adjacent sides; for instance, on Fig.29 AE
: CE = AB : BC .
Midperpendicular is a perpendicular, drawn from a middle point of a segment (side).Three midperpendiculars
of a triangle ( ABC, Fig.30 ), each drawn through the middle of its side ( points K, M, N, Fig.30 ), intersect in
one point O, which is a center of circle, circumscribed around the triangle ( circumcircle ).
In an acute-angled triangle this point lies inside of the triangle; in an obtuse-angled triangle - outside of the
triangle; in a right-angled triangle - in the middle of the hypotenuse. An orthocenter, a center of gravity, a center
of an inscribed circle and a center of a circumcircle coincide only in an equilateral triangle.
Pythagorean theorem. In a right-angled triangle a square of the hypotenuse length is equal to a sum of squares
of legs lengths.
A proof of Pythagorean theorem is clear from Fig.31. Consider a right-angled triangle ABC with legs a, b and
a hypotenuse c.
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Build the square AKMB, using hypotenuse AB as its side. Then continue sides of the right-angled triangle ABC
so, to receive the square CDEF, the side length of which is equal to a + b . Now it is clear, that an area of the
square CDEF is equal to ( a + b )². On the other hand, this area is equal to a sum of areas of four right-angled
triangles and a square AKMB, that is
c² + 4 ( ab / 2 ) = c² + 2 ab ,
hence,
c² + 2 ab = ( a + b )²,
and finally, we have:
c² = a² + b².
Relation of sides’ lengths for arbitrary triangle.
In general case ( for any triangle ) we have:
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c² = a² + b² – 2ab · cos C,
where C – an angle between sides a and b .
Parallelogram and trapezoid
Parallelogram ( ABCD, Fig.32 ) is a quadrangle, opposite sides of which are two-by-two parallel.
Any two opposite sides of a parallelogram are called bases, a distance between them is called a height ( BE,
Fig.32 ).
Properties of a parallelogram.
1. Opposite sides of a parallelogram are equal ( AB = CD, AD = BC ).
2. Opposite angles of a parallelogram are equal ( A = C, B = D ).
3. Diagonals of a parallelogram are divided in their intersection point into two
( AO = OC, BO = OD ).
4. A sum of squares of diagonals is equal to a sum of squares of four sides:
AC² + BD² = AB² + BC² + CD² + AD² .
Signs of a parallelogram.
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A quadrangle is a parallelogram, if one of the following conditions takes place:
1. Opposite sides are equal two-by-two ( AB = CD, AD = BC ).
2. Opposite angles are equal two-by-two ( A = C, B = D ).
3. Two opposite sides are equal and parallel ( AB = CD, AB || CD ).
4. Diagonals are divided in their intersection point into two ( AO = OC, BO = OD ).
Rectangle.
If one of angles of parallelogram is right, then all angles are right (why ?). This parallelogram is called a
rectangle ( Fig.33 ).
Main properties of a rectangle.
Sides of rectangle are its heights simultaneously.
Diagonals of a rectangle are equal: AC = BD.
A square of a diagonal length is equal to a sum of squares of its sides’ lengths ( see above Pythagorean theorem
):
AC² = AD² + DC².
Rhombus. If all sides of parallelogram are equal, then this parallelogram is called a rhombus ( Fig.34 ) .
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Diagonals of a rhombus are mutually perpendicular ( AC BD ) and divide its angles into two ( DCA =
BCA, ABD = CBD etc. ).
Square is a parallelogram with right angles and equal sides ( Fig.35 ). A square is a particular case of a
rectangle and a rhombus simultaneously; so, it has all their above mentioned properties.
Trapezoid is a quadrangle, two opposite sides of which are parallel (Fig.36).
Here AD || BC. Parallel sides are called bases of a trapezoid, the two others ( AB and CD ) – lateral sides. A
distance between bases (BM) is a height. The segment EF, joining midpoints E and F of the lateral sides, is
called a midline of a trapezoid.
A midline of a trapezoid is equal to a half-sum of bases:
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and parallel to them: EF || AD and EF || BC.
A trapezoid with equal lateral sides ( AB = CD ) is called an isoscelestrapezoid. In an isosceles trapezoid angles
by each base, are equal ( A = D, B = C ). A parallelogram can be considered as a particular case of
trapezoid.
Midline of a triangle is a segment, joining midpoints of lateral sides of a triangle. A midline of a triangle is
equal to half of its base and parallel to it.This property follows from the previous part, as triangle can be
considered as a limit case (“degeneration”) of a trapezoid, when one of its bases transforms to a point.
Volumes and areas of body surfaces
Designations: V – a volume; S – a base area; Slat – a lateral surface area; P – a full surface area; h – a
height; a, b, c – dimensions of a right angled parallelepiped; A – an apothem of a regular pyramid and a
regular truncated pyramid; L – a generatrix of a cone; p – a perimeter or a circumference of a base; r – a
radius of a base; d – a diameter of a base; R – a radius of a ball; D – a diameter of a ball; indices 1 and
2 are related to radii, diameters, perimeters and areas of upper and lower bases of truncated prism and
pyramid.
A prism ( right and oblique ) and a parallelepiped:
V = Sh .
A right prism:
Slat = ph .
A right angled parallelepiped:
V = abc ; P = 2 ( ab + bc + ab ) .
A cube:
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V = a ³ ; P = 6 a ² .
A pyramid ( regular and irregular ) :
A regular pyramid:
A truncated pyramid ( regular and irregular ) :
A regular truncated pyramid:
A circular cylinder ( right and oblique ):
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A round cylinder :
A circular cone ( round and oblique):
A round cone:
A truncated circular cone ( round and oblique ):
A truncated round cone:
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A sphere ( ball ):
A hemisphere:
A spherical segment:
A spherical layer:
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A spherical sector:
here h – a height of a segment, contained in the sector.
A hollow ball:
here R1 , R2 , D1 , D2 – radii and diameters of external and internal spherical surfaces correspondingly.
Circle
A circle ( Fig.1 ) is a locus of points, equidistant from the given point О, called a center of circle, at
the distance R. A number R > 0 is called a radius of circle.
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Math III - 22 -
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An equation of circle of radius R with a center in a point О ( х0 , у 0 ) is:
( х – х0 ) 2
+ ( у – у 0 ) 2
= R 2
.
If a center of the circle coincides with the origin of coordinates, then an equation of circle becomes:
х 2
+ у 2
= R2
.
Let Р ( х1 , у 1 ) be a point of the circle ( Fig.1 ), then an equation of tangent line to circle in the given point is:
( х1 – х0 ) ( х – х0 ) + ( у1 – у 0 ) ( у – у 0 ) = R2
.
A tangency condition of a straight line y = m x + k and a circle х 2
+ у 2
= R2
:
k 2
/ ( 1 + m 2
) = R2
.
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